# methods of interpolation

## methods of interpolation

السلام عليكم ورحمة الله اريد لو سمحتم methods of interpolation in gis and geodesy

بالانجليزيه لوسمحتم وشكرا جزيلا

بالانجليزيه لوسمحتم وشكرا جزيلا

**mohamed saif**- عدد الرسائل : 2

تاريخ التسجيل : 15/08/2012

## رد: methods of interpolation

mohamed saif كتب:السلام عليكم ورحمة الله اريد لو سمحتم methods of interpolation in gis and geodesy بالانجليزيه لوسمحتم وشكرا جزيلا

السلام عليكم

المكتبة الرقمية للمنتدي (أنظر الرابط التالي) تحتوي عدة كتب بالانجليزية عن الجيوديسيا وستجد بها موضوعات عدة عن تطبيقات interpolation في عدد من أقسام الجيوديسيا:

http://surveying.ahlamontada.com/t167-topic

**وكمثال (سريع) فأن الطرق المستخدمة في برنامج السيرفر - أنظر صفحة المساعدة help ان كان لديك هذا البرنامج - تشمل الطرق التالية و نبذة عن طل طريقة:**

Kriging and Variograms

The kriging algorithm incorporates four essential details:

When computing the interpolation weights, the algorithm considers the spacing between the point to be interpolated and the data locations. The algorithm considers the inter-data spacings as well. This allows for declustering.

When computing the interpolation weights, the algorithm considers the inherent length scale of the data. For example, the topography in Kansas varies much more slowly in space than does the topography in central Colorado. Consider two observed elevations separated by five miles. In Kansas it would be reasonable to assume a linear variation between these two observations, while in the Colorado Rockies such an assumed linear variation would be unrealistic. The algorithm adjusts the interpolation weights accordingly.

When computing the interpolation weights, the algorithm considers the inherent trustworthiness of the data. If the data measurements are exceedingly precise and accurate, the interpolated surface goes through each and every observed value. If the data measurements are suspect, the interpolated surface may not go through an observed value, especially if a particular value is in stark disagreement with neighboring observed values. This is an issue of data repeatability.

Natural phenomena are created by physical processes. Often these physical processes have preferred orientations. For example, at the mouth of a river the coarse material settles out fastest, while the finer material takes longer to settle. Thus, the closer one is to the shoreline the coarser the sediments, while the further from the shoreline the finer the sediments. When computing the interpolation weights, the algorithm incorporates this natural anisotropy. When interpolating at a point, an observation 100 meters away but in a direction parallel to the shoreline is more likely to be similar to the value at the interpolation point than is an equidistant observation in a direction perpendicular to the shoreline.

Inverse Distance to a Power

The Inverse Distance to a Power gridding method is a weighted average interpolator, and can be either an exact or a smoothing interpolator.

With Inverse Distance to a Power, data are weighted during interpolation such that the influence of one point relative to another declines with distance from the grid node. Weighting is assigned to data through the use of a weighting power that controls how the weighting factors drop off as distance from a grid node increases. The greater the weighting power, the less effect points far from the grid node have during interpolation. As the power increases, the grid node value approaches the value of the nearest point. For a smaller power, the weights are more evenly distributed among the neighboring data points.

Normally, Inverse Distance to a Power behaves as an exact interpolator. When calculating a grid node, the weights assigned to the data points are fractions, and the sum of all the weights are equal to 1.0. When a particular observation is coincident with a grid node, the distance between that observation and the grid node is 0.0, and that observation is given a weight of 1.0, while all other observations are given weights of 0.0. Thus, the grid node is assigned the value of the coincident observation. The Smoothing parameter is a mechanism for buffering this behavior. When you assign a non-zero Smoothing parameter, no point is given an overwhelming weight so that no point is given a weighting factor equal to 1.0.

One of the characteristics of Inverse Distance to a Power is the generation of " bull's-eyes" surrounding the position of observations within the gridded area. You can assign a smoothing parameter during Inverse Distance to a Power to reduce the "bull's-eye" effect by smoothing the interpolated grid.

Inverse Distance to a Power is a very fast method for gridding. With less than 500 points, you can use the All Data search type and gridding proceeds rapidly.

Triangulation with Linear Interpolation

The Triangulation with Linear Interpolation method in Surfer uses the optimal Delaunay triangulation. The algorithm creates triangles by drawing lines between data points. The original points are connected in such a way that no triangle edges are intersected by other triangles. The result is a patchwork of triangular faces over the extent of the grid. This method is an exact interpolator.

Each triangle defines a plane over the grid nodes lying within the triangle, with the tilt and elevation of the triangle determined by the three original data points defining the triangle. All grid nodes within a given triangle are defined by the triangular surface. Because the original data are used to define the triangles, the data are honored very closely.

Triangulation with Linear Interpolation works best when your data are evenly distributed over the grid area. Data sets that contain sparse areas result in distinct triangular facets on the map.

Nearest Neighbor

The Nearest Neighbor gridding method assigns the value of the nearest point to each grid node. This method is useful when data are already evenly spaced, but need to be converted to a Surfer grid file. Alternatively, in cases where the data are nearly on a grid with only a few missing values, this method is effective for filling in the holes in the data.

Sometimes with nearly complete grids of data, there are areas of missing data that you want to exclude from the grid file. In this case, you can set the Search Ellipse to a value so the areas of no data are assigned the blanking value in the grid file. By setting the search ellipse radii to values less than the distance between data values in your file, the blanking value is assigned at all grid nodes where data values do not exist.

When you use the Nearest Neighbor method to convert regularly spaced XYZ data to a grid file, you can set the grid spacing equal to the spacing between data points in the file. Refer to Producing a Grid File from a Regular Array of XYZ Data for the procedure of converting regularly spaced XYZ data into a Surfer grid file.

Modified Shepard's Method

Modified Shepard's Method uses an inverse distance weighted least squares method. As such, Modified Shepard's Method is similar to the Inverse Distance to a Power interpolator, but the use of local least squares eliminates or reduces the "bull's-eye" appearance of the generated contours. Modified Shepard's Method can be either an exact or a smoothing interpolator.

The Surfer algorithm implements Franke and Nielson's (1980) Modified Quadratic Shepard's Method with a full sector search as described in Renka (1988). (Surfer 6 was based upon Franke and Nielson, not Renka.)

Natural Neighbor

The Natural Neighbor gridding method is quite popular in some fields. What is Natural Neighbor interpolation? Consider a set of Thiessen polygons (the dual of a Delaunay triangulation). If a new point (target) were added to the data set, these Thiessen polygons would be modified. In fact, some of the polygons would shrink in size, while none would increase in size. The area associated with the target's Thiessen polygon that was taken from an existing polygon is called the "borrowed area." The Natural Neighbor interpolation algorithm uses a weighted average of the neighboring observations, where the weights are proportional to the "borrowed area."

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